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Computational Electromagnetics for RF and Microwave Engineering
Preface to the second edition page xv
Preface to the first edition xvii
Acknowledgements xxi
To the reader xxiii
List of notation xxiv
1 An overview of computational electromagnetics for RF and microwave
applications 1
1.1 Introduction 1
1.2 Full-wave CEM techniques 3
1.3 The method of moments (MoM) 8
1.4 The finite difference time domain (FDTD) method 10
1.5 The finite element method (FEM) 13
1.6 Other methods 16
1.6.1 Transmission line matrix (TLM) method 16
1.6.2 The method of lines (MoL) 17
1.6.3 The generalized multipole technique (GMT) 17
1.7 The CEM modelling process 17
1.8 Verification and validation 19
1.8.1 An example: a frequency selective surface 20
1.9 Convergence and extrapolation 23
1.10 Extending the limits of full-wave CEM methods 24
1.11 CEM: the future 25
1.12 A “road map” of this book 28
References 29
2 The finite difference time domain method: a one-dimensional introduction 32
David B. Davidson and James T. Aberle
2.1 Introduction 32
2.2 An overview of finite differences 33
2.2.1 Partial differential equations 33
2.2.2 The basic solution procedure 34
2.2.3 Approximating derivatives using finite differences 34
viii Contents
2.3 A very brief history of the FDTD 36
2.4 A one-dimensional introduction to the FDTD 37
2.4.1 A one-dimensional model problem: a lossless transmission line 37
2.4.2 FDTD solution of the one-dimensional lossless transmission line
problem 40
2.4.3 Accuracy, convergence, consistency and stability of the method 46
2.5 Obtaining wideband data using the FDTD 52
2.5.1 The Gaussian pulse 52
2.5.2 The Gaussian derivative pulse 54
2.5.3 A polynomial pulse 54
2.5.4 The 1D transmission line revisited from a wideband perspective 57
2.5.5 Estimating the Fourier transform 60
2.5.6 Simulation using Gaussian and Gaussian derivative pulses 62
2.6 Numerical dispersion in FDTD simulations 64
2.6.1 Dispersion 64
2.6.2 Derivation of the dispersion equation 66
2.6.3 Some closing comments on dispersion in FDTD grids 67
2.7 The Courant stability criterion derived by von Neumann analysis 69
2.8 Conclusion 71
References 71
Problems and assignments 72
3 The finite difference time domain method in two and three dimensions 74
3.1 Introduction 74
3.2 The 2D FDTD algorithm 74
3.2.1 Electromagnetic scattering problems 75
3.2.2 The TEz formulation 75
3.2.3 Including a source: the scattered/total field formulation 79
3.2.4 Meshing the scatterer 81
3.2.5 Absorbing boundary conditions 82
3.2.6 Developing the simulator 84
3.2.7 FDTD analysis of TE scattering from a PEC cylinder 91
3.2.8 Computational aspects 96
3.3 The PML absorbing boundary condition 97
3.3.1 An historical perspective 97
3.3.2 A numerical absorber – pre-Berenger 99
3.3.3 Berenger’s split field PML formulation 101
3.3.4 The FDTD update equations for a PML 102
3.3.5 PML implementation issues 104
3.3.6 Results for a split field PML 105
3.3.7 Drawbacks of the Berenger PML 106
3.3.8 Uniaxial absorber theory 107
3.3.9 Stretched coordinate theory 108
Contents ix
3.3.10 Further reading on PMLs 108
3.3.11 Conclusions on the PML 109
3.4 The 3D FDTD algorithm 109
3.4.1 The Yee cell in 3D 110
3.4.2 An application: determining the resonant frequencies of a
PEC cavity 114
3.4.3 Dispersion in two and three dimensions 115
3.5 Commercial implementations 117
3.5.1 An introductory example – a waveguide “through” 118
3.5.2 A waveguide filter 120
3.5.3 A microstrip patch antenna 121
3.6 Further reading 124
3.7 Conclusions 125
References 126
Problems and assignments 127
4 A one-dimensional introduction to the method of moments: modelling thin wires
and infinite cylinders 130
4.1 Introduction 130
4.2 An electrostatic example 131
4.2.1 Some simplifying approximations 132
4.2.2 Approximating the charge 133
4.2.3 Collocation 134
4.2.4 Solving the system of linear equations 135
4.2.5 Results and discussion 136
4.3 Thin-wire electrodynamics and the MoM 137
4.3.1 The electrically thin dipole 137
4.3.2 A caveat regarding thin-wire formulations 144
4.4 More on basis functions 144
4.4.1 The numerical electromagnetic code (NEC) – method of moments 144
4.4.2 NEC basis functions 145
4.4.3 Piecewise linear basis functions 147
4.4.4 Junction treatments with piecewise linear basis functions 147
4.5 The method of weighted residuals 150
4.6 Scattering from infinite cylinders 152
4.6.1 General derivation of surface integral equation operators 153
4.6.2 The EFIE for TM scattering 154
4.6.3 MoM solution of EFIE for TM scattering 155
4.6.4 Coding in MATLAB for right circular PEC cylinder 157
4.6.5 Post-processing: echo width and radar cross-section 157
4.6.6 Discussion, and the Fredholm alternative 159
4.7 Further reading 160
4.8 Conclusions 162
x Contents
References 162
Problems and assignments 164
5 The application of the FEKO and NEC-2 codes to thin-wire antenna modelling 166
5.1 Introduction 166
5.2 An introductory example: the dipole 168
5.3 A wire antenna array: the Yagi–Uda antenna 172
5.4 A log-periodic antenna 177
5.5 An axial mode helix antenna 185
5.6 A Wu–King loaded dipole 193
5.7 Conclusions 199
References 199
6 The method of moments for surface modelling 201
6.1 Electric and magnetic field integral equations 201
6.2 The Rao–Wilton–Glisson (RWG) element 203
6.3 A mixed potential electric field integral equation for electromagnetic
scattering by surfaces of arbitrary shape 206
6.3.1 The electric field integral equation (EFIE) 206
6.3.2 The RWG basis function revisited 207
6.3.3 The MoM formulation 208
6.3.4 Derivation of the matrix entries 210
6.3.5 Numerical approximation of the matrix entries 211
6.3.6 Coding issues 214
6.3.7 Verification 215
6.3.8 Discussion 217
6.4 Some examples of surface modelling 218
6.4.1 Scattering from a sphere 218
6.4.2 The analytical solution 222
6.5 Modelling homogeneous material bodies using equivalent currents 224
6.6 Scattering from a dielectric sphere 226
6.7 Computational implications of surface and volume modelling with
the MoM 228
6.8 Hybrid MoM/asymptotic techniques for large problems 230
6.8.1 Introduction 230
6.8.2 Moment method/asymptotic hybrids 231
6.8.3 Physical optics and MoM hybridization 231
6.8.4 A FEKO example using the MoM/PO hybrid 234
6.9 Other approaches for the solution of electromagnetically large problems 237
6.9.1 Background 237
6.9.2 High-performance computing 238
6.9.3 FFT-based methods 248
6.9.4 The fast multipole method 251
Contents xi
6.10 Further reading 258
6.11 Concluding comments 260
References 260
Problem 263
7 The method of moments and stratified media: theory 264
7.1 Introduction 264
7.2 Dyadic Green functions: some introductory notes 264
7.3 A static example of a stratified medium problem: the grounded
dielectric slab 266
7.4 The Sommerfeld potentials 269
7.4.1 A brief revision of potential theory 269
7.4.2 The Sommerfeld potentials 270
7.4.3 An example: derivation of Gxx
A for single-layer microstrip 273
7.4.4 The scalar potential and the mixed potential integral equation 276
7.4.5 Surface waves 277
7.5 Evaluating the Sommerfeld integrals 278
7.5.1 Approximate evaluation of the Sommerfeld integrals 278
7.5.2 Numerical integration in the spectral domain 279
7.5.3 Locating the pole 287
7.5.4 General source locations 288
7.5.5 Some results for the Sommerfeld potentials 289
7.6 MoM solution using the Sommerfeld potentials 289
7.7 Further reading 297
References 298
Assignments 299
8 The method of moments and stratified media: practical applications of a
commercial code 300
8.1 Printed antenna and microstrip technology: a brief review 300
8.2 A simple patch antenna 301
8.3 Mutual coupling between microstrip antennas 303
8.4 An array with “scan blindness” 308
8.5 A concluding discussion of stratified media formulations 314
References 315
9 A one-dimensional introduction to the finite element method 317
9.1 Introduction 317
9.2 The variational boundary value problem: the transmission line
problem revisited 318
9.2.1 The model problem 319
9.2.2 The equivalent variational functional 320
xii Contents
9.2.3 The finite element approximation of the functional 321
9.2.4 Evaluating the elemental matrices 323
9.2.5 Assembling the system 325
9.2.6 Rendering the functional stationary and solving the problem 327
9.2.7 Coding the FEM 328
9.2.8 Results and rate of convergence 329
9.3 Improving and generalizing the FEM solution 331
9.3.1 Higher-order elements 331
9.3.2 More general boundary conditions 337
9.4 Further reading 339
9.5 Conclusions 340
References 340
Problems and assignments 341
10 The finite element method in two dimensions: scalar and vector elements 342
10.1 Introduction 342
10.2 Finite element solution of the Laplace equation in two dimensions using
scalar elements 343
10.2.1 The variational boundary value problem approach 343
10.2.2 Some practical issues: assembling the system 349
10.2.3 An application to microstrip 352
10.2.4 More on variational functionals 355
10.2.5 The Poisson equation: incorporating a source term 358
10.2.6 Discussion 358
10.3 The Galerkin (weighted residual) formulation 359
10.4 Simplex coordinates 364
10.4.1 Simplex coordinates in one, two and three dimensions 365
10.4.2 Some properties of simplex coordinates 366
10.5 The high-frequency variational functional 367
10.6 The null space of the curl operator and spurious modes 367
10.7 Vector (edge) elements 371
10.7.1 An historical perspective 371
10.7.2 Theory of vector elements 372
10.7.3 Vector elements on triangles – the Whitney element 374
10.8 Application to waveguide eigenvalue analysis 378
10.8.1 The two-dimensional variational functional for an homogeneous
waveguide 378
10.8.2 Explicit formula for the elemental matrix entries 379
10.8.3 Coding 382
10.8.4 Results 386
10.8.5 Degenerate modes 389
10.8.6 Higher-order vector elements 391
10.9 Waveguide dispersion analysis 394
Contents xiii
10.9.1 A vector formulation based on the transverse and axial field
components 394
10.9.2 The cut-off eigenanalysis formulation 396
10.9.3 Homogeneously filled guides: TE modes only 397
10.9.4 Eigensolution 398
10.9.5 Results: a half-filled dielectric loaded rectangular waveguide 398
10.9.6 Alternate formulations for inhomogeneously loaded waveguides 400
10.10 Further reading 400
10.11 Conclusions 402
References 403
Problems and assignments 406
11 The finite element method in three dimensions 407
11.1 The three-dimensional Whitney element 407
11.1.1 Explicit formula for the tetrahedral elemental matrix entries 408
11.1.2 Coding 411
11.2 Higher-order elements 415
11.2.1 Complete versus mixed-order elements 416
11.2.2 Hierarchal vector basis functions 416
11.2.3 Properties of hierarchal basis functions 419
11.2.4 Practical impact of higher-order basis functions in an FEM code 421
11.3 The FEM from the variational boundary value problem viewpoint 427
11.4 A deterministic 3D application: waveguide obstacle analysis 429
11.4.1 Introduction 429
11.4.2 The waveguide formulation 430
11.5 Application to two waveguide discontinuity problems 432
11.5.1 Application to a Magic-T 432
11.5.2 Application to a capacitive iris 436
11.6 Open-region finite element method formulations: absorbing boundary
conditions (ABCs) 441
11.6.1 Formulation in terms of the scattered field 442
11.6.2 Formulation in terms of the total field 443
11.6.3 Discussion 444
11.7 Further reading 444
11.8 Conclusions 445
References 445
Problems and assignments 448
12 A selection of more advanced topics in full-wave computational electromagnetics 451
12.1 Hybrid finite element/method of moments formulations 451
12.1.1 Introduction 451
12.1.2 Theoretical background 452
12.2 An application of the FEM/MoM hybrid – GSM base stations 454
xiv Contents
12.2.1 Applications of FEM/MoM hybrid formulations 454
12.2.2 Human exposure assessment near GSM base stations 455
12.3 Time domain FEM 457
12.3.1 Basic formulation and implementation 458
12.3.2 Preliminary results 461
12.3.3 The FDTD as a special case of the FETD 464
12.3.4 Hybrid FDTD/FETD schemes 468
12.4 Sparse matrix solvers 468
12.4.1 Profile-in skyline storage 469
12.4.2 Compressed row storage 470
12.4.3 Implementation of matrix solution using these storage schemes 471
12.4.4 Results for sparse storage schemes 471
12.5 A posteriori error estimation and adaptive meshing 473
12.5.1 Explicit, residual-based error estimators 474
12.5.2 An example of the application of an error estimator 476
12.6 Further reading and conclusions 478
References 481
Appendix A: The Whitney element 484
Appendix B: The Newmark-β time-stepping algorithm 486
References 488
Appendix C: On the convergence of the MoM 489
Reference 490
Appendix D: Useful formulas for simplex coordinates 491
Appendix E: Web resources 493
Appendix F: MATLAB files supporting this text 496
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